Maneuvering assistant for truck and trailer combinations with arbitrary trailer hitching

Transcription

1 Maneuvering assistant or truck and trailer combinations with arbitrary trailer hitching Yevgen Sklyarenko, Frank Schreiber and Walter Schumacher Abstract Stabilizing controllers can assist the driver to saely maneuver o truck and trailer combinations like the EuroCombis. This paper presents a novel nonlinear controller or combinations o a nonholonomic tractor car with two semitrailers based upon a instantaneous approximation o the vehicle kinematics, an exact inputstate linearization techniue and a modelreerence control scheme. The main advantage o the suggested controller is its universality, as it can be applied to truck and trailer combinations with an arbitrary hitching oset as well as in both driving directions. Convergence and robustness properties o the proposed controller have been validated by series o simulations and experiments with a combination, which possesses one oaxle and one onaxle joint. The presented method can also be extended to other truck and trailer combinations. Keywords truck and trailer combination, oaxle hitching, inputstate linearization, modelreerence control scheme I. INTRODUCTION Automatic and autonomous transportation systems are the objective o numerous research initiatives at present. Among those the controllability o articulated multibody vehicles plays a prominent role. The major control challenge in this type o plant arises mainly rom the nonlinear nature o their kinematics and its instability in the case o backward motion in a conjunction with the steering limitations. The introduction o the European Council Directive 96/53/EG in 996 created a legal basis within the European Union or truck and trailer combinations up to lengths o 5.5 m and up to a total weight o 60 tons. As a result o the developments in the logistics industry dierent truck and trailer conigurations emerged, which are known as EuroCombis, see Fig.. Most o those EuroCombis variants A E eature two hitching joints and can be regarded kinematically as a tractor car with two semitrailers. However, these EuroCombis dier rom each other with regard to their trailer hitching osets. This is a crucial point which leads to partially dierent behavior o the vehicles. The ability to handle systems with an arbitrary type o trailer hitching is thereore o utmost importance or designing a universal maneuvering assistant or the most relevant truck and trailer combinations with two semitrailers TTC. Through the many years o international research numerous stabilizing controllers or articulated vehicles were developed, depending on the existence and direction o the trailer hitching oset. The earlier works, based on the method o exact inputstate linearization, were limited to the case, All authors are with the Technische Universität Braunschweig, Institute o Control Engineering, HansSommerStr. 66, 3806 Braunschweig, Germany corresponding author: skliarenko@ir.ing.tubs.de Fig.. Variants o the EuroCombis [] that the hitching point was located directly on the center o the rear axle o the preceding towing unit socalled onaxle hitching [], [3], [4], [5]. It was also noted that the exact inputstate linearization is not applicable or multibody articulated vehicles with an oset in the trailer hitching socalled oaxle hitching [6], [7]. In [7] an approximation o the system with a hitching oset at the truck by an onaxle system was suggested. The approximated ghost vehicle provided the same steady state behavior and allowed an exact inputstate linearization. During pathtracking the modeling inaccuracies were considered as disturbances. Similar results were later achieved or trainlike vehicles with a hitching oset by applying the method o exact inputoutput linearization instead o inputstate linearization [8], [9]. The peculiarity o these control laws is the dependence o the choice o the guidepoint and thereore the controller structure has to be adjusted based upon the driving direction. It should be noted that most works on the control o articulated vehicles are limited to a negative hitching oset joint behind the rear axle o the preceding unit, while to the present time there are relatively ew research results available regarding the control o trainlike vehicles with positive or an arbitrary hitching oset. The reason are the mentioned diiculties to apply the method o exact linearization. In [0] this problem was avoided by conventional Jacobian linearization. Promising results provides the application o the Lyapunov method to design a controller. In [] this is shown only or a vehicle with a single semitrailer. In addition the application o known approaches to the pathtracking problem or trainlike vehicles with oaxle hitching provides uite complex control laws and complicates their practical application.

2 The measures proposed in this paper or TTC with arbitrary trailer hitching dier rom previous approaches by the underlying hypothesis that the stabilization and the pathtracking tasks can be treated separately, resulting in a cascaded control structure. The novelty o the proposed lowerlevel stabilizing control loop, which orms a maneuvering assistant, arises rom applying an instantaneous approximation o the vehicle kinematics, an exact inputstate linearization techniue and a modelreerence control scheme. This approach oers all the advantages o a cascade control structure and will allow at a later time the design o simpler control laws o a superimposed pathtracking controller. II. STABILIZING CONTROLLER A. Analysis o possible solutions The nonlinear control theory provides a variety o methods or the analysis and synthesis o nonlinear control systems. The most successul method applied to truck and trailer combinations is still the exact eedback linearization. The special attractiveness o this method is explained by the elegant way, in which the reuirements o stability and control perormance can be addressed using the wellknown methods o linear control theory to the linearized system. There are two wellknown exact linearization techniues, inputoutput linearization IOL and inputstate linearization ISL, applicable or the class o nonlinear aine in control systems. The application o IOL is usually more straightorward than that o ISL. However, a control law based on IOL is only suitable with a stable internal dynamics o the controlled system. In case o IOL o a truck and trailer combination the stability o the internal dynamics as shown in [8] depends o the direction o motion, the sign o the hitching oset and a choice o the guidepoint o the vehicle. Since the ISL produces no internal dynamics in the controlled system, it would allow to design the universal control laws valid or truck and trailer combinations with an arbitrary hitching oset as well as independent o the direction o motion. Unortunately, the analytical determination o the lat output or the nonlinear system, like a TTC, can be an unachievable task. Thereore, irregular approaches to simpliy the task may be reuired. It is wellknown, that a substitution o variables in nonlinear dierential euations o the vehicle s model provides an aine in control system without increasing the order o the system. Our investigation has shown, that a urther instantaneous approximation o a single term in the euations allows to obtain the solution o a partial dierential euation leading to a lat output o the approximated system with relative ease. Based upon this approach, the state as well the input transormation can be ound. Ater such o an approximate inputstate linearization the design o a linear state regulator or the stabilizing and tracking controller maneuvering assistant can be done. To provide robust perormance under the inluence o model inaccuracies and disturbances a modelreerence control scheme is used. B. Kinematic model o the vehicle with two semitrailers The design o a controller reuires a model o the TTC that describes the system behavior in orward and reverse direction o movements with suicient accuracy. A purely kinematic model can be established, which neglects dynamic eects e.g. inertia and slip and reduces to a planar singletrack model. The kinematic modeling leads to a much simpler and better interpretable system o dierential euations, which reproduces the processes in truck and trailer combinations with reasonable accuracy. Fig. depicts the schematic representation o a TTC. The irst joint is assumed to be an oaxle joint with an arbitrary hitching oset pictured positive while the second hitching is onaxle. The ollowing variables and parameters are illustrated in Fig. : φ 0 t steering angle φ t angle between irst semitrailer and tractor φ t angle between second and irst semitrailer λ distance traveled along the path V 0 t tractors ront axle speed along the path V t speed o irst semitrailer l 0 distance between tractors axles d hitching oset positive towards tractors ront axle l length o irst semitrailer l length o second semitrailer Fig.. l l d V Schematic o a vehicle with two semitrailers For the design o a stabilizing controller only the behavior o the internal degrees o reedom φ t and φ t have to be considered. For ease o analysis and design o the control system, the ollowing notations were introduced: state variables: x = φ, x = φ ; control variables: x 3 = φ 0, V 0 ; d model parameters: a = V 0 l, a = V 0 l 0l, a 3 = V 0 l, d a 4 = V 0 l 0l, a 5 = V 0 l 0 ; short orm o the unctions: c i = cos x i, s i = sin x i. l 0 V 0 l 0

3 Under these notations the behavior o the internal coordinates φ t und φ t without consideration o external degrees o reedom leads to a system o two nonlinear dierential euations: ẋ = a s c c 3 a s s s 3 a 3 s c 3 a 4 c s 3 ẋ = a 3 s c 3 a 4 c s 3 a 5 s 3 ISL is applicable to a class o nonlinear aine in control systems, whose origin is an euilibrium with zero control: ẋ = x g x u, x R n, u R, with x and g x smooth n vector unctions. To convert the model into, the speed o the irst semitrailer has to be used instead o the tractors ront axle speed. This is euivalent to introducing the ollowing variable substitution: Further, ater another substitution V = V 0 c c 3. 3 u = tan x 3 4 the unctions o the aine in control plant model become: [ ] [ ] x s = = 3 tan x, x 3 tan x [ ] [ ] g x s tan x g = = 4 g x c The model parameters i correspond to the parameters a i, i the speed V replaces V 0. This means, that the speed V must be kept constant by the tractor s speed control according to 3. The speed V 0 can be completely eliminated rom the dierential euations to avoid singularities at V 0 = 0, i the time derivatives are replaced by the derivatives with respect to a new independent variable λ, e.g. by means o the socalled time scaling []. C. Approximation and inputstate linearization ISL is achieved by a conversion rom a system o nonlinear dierential euations to the canonical Brunovsky orm [3]: ż = z, ż = z 3,... ż n = z n ż n = v by a linearizing eedback, consisting o: dieomorphic state transormation z x = [z x... z n x] T and input transormation u = β z v α z, with v the new linearized input. The plant in this case is covered by the eedback on theoretically all state variables. From this ollows the deinition o the eedback linearization. There are no internal dynamics, because all state variables are linearized by ISL. The principal dierence rom the conventional Jacobian linearization is that the exact linearization is not approximate but an euivalent transormation. 6 The plant, 5 is controllable in the [ vicinity o the origin, where controllability matrix Q c = ad 0 g... adn ] g has ull rank. It is easy to obtain the controllability conditions in the origin: 4 5 and 4 3 5, which can be summarized into constraints on the vehicle s geometry: d l and d l. In truck and trailer combinations the hitching oset is almost always less than the length o trailers, thereore the plant is almost always controllable in the neighborhood o the origin. For { a second order system } the involutivity condition o the set ad 0 g... adn g is always satisied. This means that a dieomorphic state transormation z x or the system, 5 theoretically exists. The irst element o the state transormation vector is ound by solving the system o n homogeneous partial dierential euations dz d x adi g = 0, i = 0... n. Unortunately, the analytic solution o the single partial dierential euation or the system, 5 z x sin x tan x 4 z x 4 5 = 0 cos x and the lat output can t be ound. I we assume that 4 is the momentary value o sin x tan x 4, then the instantaneously approximated euation z x 4 z x 4 5 cos x = 0 results, or which the expression o the lat output can easily be ound: z x = 4 5 arctan tan x x x or 5 > 4 or l > d 7 Based upon this, the next elements o the state transormation can be derived as z i = L i z = L z i, i =... n [3]: z x = s 3 tan x 3 4 s 4 c 5. 8 The calculation o the input transormation also does not cause too many diiculties α x = L n z = L z n, β x = L g L n z = L g z n [3]: α x = c 3 tan x s c 4 c 5 c 3 tan x, 9 β x = 4 c c 4 c 5 4 c 5 c 3.0 c With 5 < 4 or l < d, although this is not typical or truck and trailer combinations, the lat output or the approximated system also exists z x = arctanh 5 4 tan x x x. This leads to the same results or the unctions z x, α x and β x.

4 D. Control law Considering the substitution 4 the nonlinear control law [3] u = v α x β x = z d β x b T z d z α x = β x z d b z d z b 0 z d z α x φ 0 = arctan u can be represented in a orm o two loops, comprising the plant, see Fig. 3: linearization loop with input v according to 6, nonlinear unctions β x and arctan in the eedorward path, as well as α x in the eedback path; poleplacement loop with the linear gain b = [b 0 b ] T according to the state z = [z x z x] T with reerences z d = [z d x d z d x d ] T and z d. When choosing a stable characteristic polynomial Ns = s b s b 0 the control law will provide local stability o the closed system, asymptotic convergence to zero o the tracking error εt = z d t zt and the boundedness o all the state variables [3]. In this case transient processes o the lat output o the plant will be described by a linear transer [3]. The coeicients b 0 and b o the polynomial Ns must be chosen based on the desired perormance o the closed system as well on the perormance o the steering servo. unction Gs = Ns E. Control scheme The tracking controller reuires smooth reerence signals z d, ż d,..., z n d. By a common latnessbased techniue they are generated in z ater transormation o the reerence signals x d to z d. Additionally, to compensate tracking errors occurring under the inluence o model uncertainties and disturbances the measured system output x s has to be compared with the output o the nonlinear system model x m, calculated rom z m by the inverse transormation. Here two problems arise: the state transormation is only known or the approximated system, additionally the analytic solution o 7 and 8 is necessary whose derivation is a hard task. A modelreerence control scheme was chosen or this application, comprising a reerencemodel control loop and a disturbance control loop, see Fig. 4. The reerencemodel control loop contains the instantaneously approximated nonlinear system model, which is exactly eedback linearized by 9 and 0. The output o the modelreerence control loop is x m, which can thus be obtained without an inverse transormation. The state vector z m serves as reerence vector or the disturbance control loop, while the linearized control v m is passed as a eedorward signal into the linearized part o the disturbance control loop. Due to the eedorward action the stability o the control loops is independent o each other and their controllers can be designed independently or their speciic tasks. As the model control loop is disturbance ree, its controller was designed to provide good tracking perormance by deining the polynomial N m s as a second order binomial polynomial. In the absence o disturbances the eedorward control rom the model loop would drive the output o the real plant x s along the trajectories o the instantaneous approximated model x m, because both models have the same steady state in x. Since the main tracking control is perormed in the model loop, the disturbance controller can be designed or good disturbance rejection as its role is to suppress disturbances acting rom the environment or as the result o approximation errors. Thereore, the polynomial N s s was parameterized using maximal possible dynamics under the given dynamics o the steering servo. However both models, the ull model and the approximated model, possess the same steady state in in x, but due to the approximated nature o the state transormation system model, the steady states in z are not eual. To adjust or the incorrect state transormation, the ollowing method was applied. From 8 the value or x d is determined or z d = 0 and the reerence value o x d. This calculation could have been evaluated analytically with much eort, but we have chosen to solve it numerically in closed loop in the reerence generator, see Fig. 4. Based upon the value o x d and the reerence value o x d the value o z d was calculated according to 7. The reerence vector z d, which was derived this way, is used as reerence or the reerencemodel control loop. Actually, this ensures the compliance with the steady state compatibility condition using control engineering techniues, similar to the geometric approach taken by Bolzern et al. [8]. To guarantee correct steady states in spite o parametric uncertainties o the plant model, in the last step o designing the modelreerence control scheme an overlaying PIcontroller was introduced at the input o the disturbance control loop, see Fig. 4. F. Simulation results The computer simulation was completed or the demonstrator vehicle at the Institute o Control Engineering with the ollowing parameters: l 0 = 0.84m, l = 0.6m, l = 0.9m, d = 0.4m, φ 0 max = π/6 and the time constant o the steering servo T servo = 0.s. As the polynomial N m s in the control law o the reerencemodel control loop a secondorder polynomial was used with the characteristic reuency ω 0m =, 35s m/s and b m = ω 0m, b 0m = ω 0m. The corresponding parameters or the disturbance control loop are given ω 0s = T servo m/s and b s = ω 0s D, b 0s = ω 0s, with normalized damping actor D. To demonstrate the robustness o the approach, vehicle parameters with a random variation o up to ±0% were used in the designed nonlinear controller. Simulations o the reversing process have shown the high uality o the control or TTCs with negative and positive hitching oset, see the system states and the control eorts in Fig. 5 and Fig. 6.

5 .. linear regulator z d t vt linearized plan x ut arctan 0 t Plant aine in control plant z d t b T t x linearization loop poleplacement loop zt State transorm. xt=[ t t] T Fig. 3. Signal low in the exact linearization based controller reerencemodel control loop d steady state calculation x d state transorm. z d b T m v m servo model linearized virtual plant 0m x m z m disturbance control loop x m x s PIcontroller Dz z m T b s v m v s 0s linearized real plant x s z s Fig. 4. Nonlinear reerencemodel controller scheme At λ = 0 the system was subjected to a stepwise change o the reerence or angle φ, which deines the desired path curvature o the combination during reversing. The reerence value φ d is continually calculated rom the steady state compatibility condition. As can be seen in Fig. 5 let and Fig. 6 let, the desired value φ d is met well in the approximated and the ull plant model. Due to the deviation between the approximated and the ull plant model the desired value φ d is achieved with diering values or φ m and φ s. For this reason, the corresponding steering angles φ 0m and φ 0s are also not eual, see Fig. 5 right and Fig. 6 right. At λ = 6m a disturbance is included in orm o a stepwise reduction o the steering angle by 0% o the maximal value, which simulates a lateral slip eect. It can be seen that this disturbance is suppressed eectively. The intricate task o TTC reversing with a positive oset, which is more complicated under exact linearization, has been successully controlled using the same control algorithm with a comparably high uality perormance, see Fig. 6. Note the same steady state steering angle or the negative and the positive oset case, that actually just corresponds to the Ackermann steering angle. Although a part o the controller relies on a simpliication o the nonlinear plant model, it was shown that the designed controller allows to eectively steer the motion o TTC with an arbitrary oset o the irst joint and in both directions o motion. The presented approach was also implemented and successully validated on the demonstrator vehicle, see Fig. 7 and webpage [4]. III. CONCLUSION This paper presented the design o a stabilization and tracking controller or a tractor car with two semitrailers and oaxle trailer hitching, whose kinematic model is not exactly inputstate linearizable. The designed nonlinear control law is based on inputstate linearization o the instantaneously approximated kinematic model, which can be exactly linearized and whose behavior provides an accurate perormance. Despite the eorts o simpliication, the designed nonlinear stabilization and tracking law is universal regarding the hitching oset and able to prevent the loss o stability in the closed system while reversing. The presented approach can also be applied to other types o truck and trailer combinations.

6 Fig. 5. State trajectories let and control eorts right in plant and model loop o a system with negative hitching oset d = 0.4m as a conseuence o a step input in the reerence and in the disturbance signal Fig. 6. State trajectories let and control eorts right in plant and model loop o a system with positive hitching oset d = 0.m as a conseuence o a step input in the reerence and in the disturbance signal Fig. 7. Vehicle used to validate the designed control algorithms REFERENCES [] T. Fleischmann. Truck trailer.jpg. trailer.jpg. [] J.P. Laumond, Controllability o a multibody mobile robot, Robotics and Automation, IEEE Transactions on, vol. 9, no. 6, pp , Dec [3] O. Sordalen and K. Wichlund, Exponential stabilization o a car with n trailers, in Decision and Control, 993., Proceedings o the 3nd IEEE Conerence on, Dec. 993, pp vol.. [4] I. Kolmanovsky and N. McClamroch, Developments in nonholonomic control problems, Control Systems Magazine, IEEE, vol. 5, no. 6, pp. 0 36, Dec [5] D. Tilbury, R. Murray, and S. Shankar Sastry, Trajectory generation or the ntrailer problem using goursat normal orm, Automatic Control, IEEE Transactions on, vol. 40, no. 5, pp , May 995. [6] R. M. Murray, Nilpotent bases or a class o nonintegrable distributions with applications to trajectory generation or nonholonomic systems, Mathematics o Control, Signals, and Systems MCSS, vol. 7, pp , 994. [7] P. Bolzern, R. DeSantis, A. Locatelli, and D. Masciocchi, Pathtracking or articulated vehicles with oaxle hitching, Control Systems Technology, IEEE Transactions on, vol. 6, no. 4, pp , July 998. [8] P. Bolzern, R. M. DeSantis, and A. Locatelli, An inputoutput linearization approach to the control o an nbody articulated vehicle, Journal o Dynamic Systems, Measurement, and Control, vol. 3, no. 3, pp , 00. [9] C. Altaini, Path ollowing with reduced otracking or multibody wheeled vehicles, Control Systems Technology, IEEE Transactions on, vol., no. 4, pp , 003. [0] R. DeSantis, J. Bourgeot, J. Todeschi, and R. Hurteau, Pathtracking or tractortrailers with hitching o both the onaxle and the oaxle kind, in Intelligent Control, 00. Proceedings o the 00 IEEE International Symposium on, 00, pp. 06. [] A. Astoli, P. Bolzern, and A. Locatelli, Pathtracking o a tractortrailer vehicle along rectilinear and circular paths: A Lyapunovbased approach, Robotics and Automation, IEEE Transactions on, vol. 0, no., pp , 004. [] M. Sampei, T. Tamura, T. Kobayashi, and N. Shibui, Arbitrary path tracking control o articulated vehicles using nonlinear control theory, Control Systems Technology, IEEE Transactions on, vol. 3, no., pp. 5 3, mar 995. [3] J.J. B. Slotine and W. Li, Applied nonlinear control. Prentice Hall, 99. [4] Y. Sklyarenko. Institute o control engineering. maneuvering assistant.